Geometric meaning of solutions to a linear equation in 2D

Learning Objectives:
1) Sketch the solutions to a linear equation in 2D
2) Determine whether a linear equation has 0 or infinitely many solutions in 2D

This video is part of a Linear Algebra course taught at the University of Cincinnati.

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[dlambert]

Nice video.

I feel compelled to write out the solution set for "ax + by = c".
*case 1:* b ≠ 0 ---> y = -a/b*x + c/b
We can write the general solution as (x , -a/b*x + c/b).
The parametric form is t (1, -a/b) + (0, c/b).
The graph is a line.

*case 2:* a ≠ 0 ---> x = -b/ay + c/a
We can write the general solution as (-b/a*y + c/a , y)
The parametric form is t (-b/a, 1) + (c/a, 0).
The graph is a line.

*case 3:* a=0, b=0, c=0
This has infinite solutions (x, y) since there are two free variables.
The graph is the entire plane.

*case 4:* a=0, b=0, c≠0.
This has no solution.
The graph is empty.

So there are 2 possibilities for the number of solutions of a single 2 variable linear equation. Infinite, or none.

[mineakikajitori]

If a=b=c=0, then the solution set of the equation ax+by=c is the xy-plane.